A particle can move along only an x axis, where conservative...

A particle can move along only an $x$ axis, where conservative forces act on it (Fig. 8 - 66 and the following table). The particle is released at $x=5.00 \mathrm{~m}$ with a kinetic energy of $K=14.0 \mathrm{~J}$ and a potential energy of $U=0 .$ If its motion is in the negative direction of the $x$ axis, what are its (a) $K$ and (b) $U$ at $x=2.00 \mathrm{~m}$ and its (c) $K$ and $($ d) $U$ at $x=0 ?$ If its motion is in the positive direction of the $x$ axis, what are its (e) $K$ and (f) $U$ at $x=11.0 \mathrm{~m}$, its (g) $K$ and (h) $U$ at $x=12.0 \mathrm{~m},$ and its $(\mathrm{i}) K$ and (j) $U$ at $x=13.0 \mathrm{~m} ?$ (k) Plot $U(x)$ versus $x$ for the range $x=0$ to $x=13.0 \mathrm{~m}$

Key Concepts

Energy Diagrams in One-Dimensional Motion

Plotting the potential energy as a function of position provides a visual representation of the system's energy landscape. Such diagrams indicate regions where the particle’s kinetic energy (and therefore its speed) is high, the turning points, and how forces vary with position, all of which aid in analyzing and predicting the motion of the particle.

Kinetic and Potential Energy

Kinetic energy is the energy associated with the motion of a particle, while potential energy is associated with the position of the particle within a force field. In problems of one-dimensional motion under conservative forces, an initial sum of these energies will remain constant, and the conversion between them determines the particle's speed and permissible positions.

Reference Point for Potential Energy

Potential energy is defined relative to an arbitrary reference level. Choosing a specific point as having zero potential energy simplifies calculations by allowing for straightforward computation of changes in energy as the particle moves, even though the absolute value may be shifted by an arbitrary constant.

Conservation of Mechanical Energy

This concept states that in the absence of non-conservative forces (like friction), the total mechanical energy (the sum of kinetic and potential energies) of a system remains constant. It enables one to relate changes in potential energy to changes in kinetic energy as a particle moves under the influence of conservative forces.

Conservative Forces

Conservative forces are forces for which the work done in moving a particle between two points is independent of the path taken. The energy lost or gained as potential energy in these fields can be completely converted to kinetic energy, which is key to solving problems involving energy transformation in one-dimensional motion.

Transcript

00:01 So, in the first sub part of the question, we need to calculate the work done from x equals to 3 meters to x equals to 2 meters.

00:09 So work done for the first sub part will be work done equals f2 multiplied to the distance.

00:21 That is equal to 5 newton multiplied to minus 1 .00 meters because the body is traveling in the negative x direction.

00:34 And the force is in the positive x direction so the force and the motion of the body are in the opposite directions so the work done will be negative that is equals to minus 5 joules in the second subpart it has asked us to calculate the kinetic energy at x equals to 2 meters we know that the maximum energy that is given to us is 14 jules so if it is the maximum energy kinetic energy at x equals to equals to 2 meter will be e max minus u2 that is equal you 2 is equal to minus 5 jules because the work done by force f2 from x equals to 3 to x equals to 2 will be converted to potential energy of the body so e max equals 14 joules u2 equals 5 so the kinetic energy at x equals to 2 meters comes out to be 9 joules now the heart support it has asked us to calculate the potential energy at x equals to 0.

01:48 So, the potential energy at x equals to 0 will be 6 joules plus u2.

01:57 That is equal to 6 plus 5 joules that equals 11 joules.

02:08 So where did we get this 6 joules from? we got this from the work done by force f1.

02:14 That is f1 multiplied to delta x equals 3 newton multiplied to minus 2 meter which is equal to minus 6 joules as we know that the work done by a conservative force equals negative of the change in potential energy so negative of minus 6 will be positive 6 so this is how we got 6 joules here so 6 joules plus u2 equals 6 plus 5 joules that is equals to 11 joules now we got our potential energy at x equals to 0.

02:50 Now in the fourth sub part it has asked us to calculate the kinetic energy at x equals to 0 and that will be equal to e max minus u .0 that is 14 minus 11 joules which equals 3 jules.

03:09 In the e sub part we need to calculate the potential energy at x equals to 11 meters.

03:16 So, the work done from x equals to 8, the work done from x equals to 8 meter to x equals to 11 meter is equal to w equals to f3 multiplied to delta x.

03:37 That is equals to minus 4 newton multiplied to 3 meters.

03:42 That equals minus 12 joules.

03:44 So the potential energy at x equals to 11 meter is equal to the potential energy at x equals to 11 meter is equal to the potential energy at x equals to 11 meters is equal to the positive of this the positive of the work done that is plus 12 joules now similarly we need to find the kinetic energy the kinetic energy at x equals to 11 meters so k 11 that i have used to denote kinetic energy at x equals to 11 meters is equals to e max minus u11 that is the potential energy at x equals to 11 meters so that comes out to be 14 minus 12 joules that is equal to 2 joules of the g subpart it asks us to calculate the potential energy at x equals to 12 meters so the work done by force f4 is f4 multiplied to delta x so that is equal to minus 1 newton multiplied to 1 meters because delta x there is 1 meters so the potential as yet x equals to sorry the work done by force f4 will be minus 1 joules so the potential energy at x equals to 12 meters will be u12 that is equal to 1 jule the the the negative of this work okay the negative of this work will give us the change in potential energy that is 1 jule plus u11 that is the potential energy at x equals to 11 meters so that comes out to be 1 plus 12 jules that equals 13 joules...

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